Application problem with a linear function: Finding a coordinate given two points

Question

The monthly cost (in dollars) of water use is a linear function of the amount of water used (in hundreds of cubic feet, HCF). The cost for using 16 HCF of water is $34.31, and the cost for using 27 HCF is $52.46. What is the cost for using 25 HCF of water?

Answer 1

So in order to solve this problem you have to determine what the price of water is as a function of HCF at 25. Start by assigning your variables by determining which one is dependent (y) on the other (x). Then calculate the average rate of change (slope) between the two points using the equation:

(y₂-y₁)/(x₂-x₁) = m

With the slope now calculated we have to determine what the y-intercept is. This can be done by plugging in some known values and solving for b (the y intercept). You can use the values from x₁ and y_1.

y₁ = mx₁+ b

b = Starting Price

Now that we have determined the equation we can solve for the price at 25 HCF.

y = m(25) + b

y = Price at 25 HCF

Hope this helps!

Answer 2

To begin this question, remember the basic form of a linear function, y=mx+b. Then, assign the variables y and x to the information given in your problem! Think about the independent variable (x) and the dependent variable (y). Does the amount of water impact the price you pay, or does the money impact how much water is used? If you think about it logically, the first proposition is the most likely. So if your x = amount of water used, and your y = the monthly cost, then you already have the information to make two points! The first would be that the cost of using 16 HCF is $34.31, and using our x and y values, we can make this coordinate point (16, 34.31). Do the exact same thing with the next piece of information! 27 HCF costs $52.46, which makes the point (27, 52.46). Now that you have those points, you can use the formula  (Δy)/ (Δx), change in y - change in x. This will give you the 'm' (slope) of your line! Now you've created an equation for a line in the form y=mx+b, so all you have to do to find the cost of using 25 HCF of water is to plug that into the equation you've just made. Let me know if you have any more questions!